S† (inverse phase) gate

Definition

$$S^\dagger = \begin{pmatrix} 1 & 0 \\ 0 & -i \end{pmatrix}$$

$S^\dagger$ is the conjugate transpose of $S = \mathrm{diag}(1,i)$, and since $S$ is unitary, $S^\dagger$ is also its inverse: $SS^\dagger = S^\dagger S = I$.

Action on basis states

$$S^\dagger|0\rangle = |0\rangle, \qquad S^\dagger|1\rangle = -i|1\rangle$$

Applied to $|+\rangle = \tfrac{1}{\sqrt2}(|0\rangle+|1\rangle)$, it produces $\tfrac{1}{\sqrt2}(|0\rangle - i|1\rangle) = |{-i}\rangle$, the state at $-\pi/2$ on the Bloch-sphere equator, mirroring how $S|+\rangle = |i\rangle$ lands at $+\pi/2$.

Key identities

• $S^\dagger = P(-\pi/2) = R_Z(-\pi/2)$ up to global phase, so $S^\dagger$ is the $\lambda=-\pi/2$ special case of the general phase gate, symmetric with $S=P(\pi/2)$.
• $S^\dagger = S^3$ since $S^4=I$: three applications of $S$ equal one $S^\dagger$, useful when a compiler's native gate set only exposes $S$.
• $S^\dagger \cdot T^2 = I$ because $T^2=S$, so $S^\dagger$ and two $T$ gates cancel exactly.
• Like $S$, $S^\dagger$ is Clifford: conjugating a Pauli operator by $S^\dagger$ returns a Pauli operator, e.g. $S^\dagger X S = -Y$.